The Hilbert matrix done right

TL;DR

This paper provides simplified proofs of classical spectral results for the Hilbert matrix, including its spectrum being [0, π] with no eigenvalues, using the Mehler-Fock transform.

Contribution

It introduces straightforward proofs of Magnus and Hill's results on the Hilbert matrix's spectral properties, utilizing the Mehler-Fock transform.

Findings

Spectrum of Hilbert matrix is [0, π]

Hilbert matrix has no eigenvalues

Characterization of eigenfunctions and eigenvalues

Abstract

We give very simple proofs of the classical results of Magnus and Hill on the spectral properties of the Hilbert matrix $$ H = \left ( {1 \over i+j+ 1 } \right )_{i,j\geq 0} $$ which defines a bounded linear operator on the sequence space $\ell^2$. In particular, we use the Mehler-Fock transform to find the spectrum and the latent eigenfunctions of the Hilbert matrix, that is, we show that the spectrum of $H$ is $[0,\pi]$ with no eigenvalues (Magnus' result) and describe all complex sequences $x$ such that $Hx=\mu x$ for some complex number $\mu$ (Hill's result).